Robin Pemantle's Lecture Notes Cornell Summer School In

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i Robin Pemantle’s Lecture Notes

Cornell Summer School in Probability 6 July – 17 July, 2009 Contents

1 Fourier-Laplace integrals 1-1

1.1Laplace-typeintegralsinonevariable...... 1-2

1.2Laplace-typeintegralsinseveralvariables...... 1-4

1.3Complexphases...... 1-7

2 Generating functions 2-1

2.1Formalpowerseriesandcombinatorialenumeration...... 2-1

2.2Applicationsinprobability...... 2-5

2.3Coeﬃcientextractionandtransfertheorems...... 2-8

3LLDandLCLT 3-1

3.1Thesmoothpointtheorem...... 3-2

3.2Locallargedeviations...... 3-4

3.3LocalCLT...... 3-7

3.4Smoothpointapplications...... 3-8

5 Proofs 5-1

5.1Firstproof...... 5-1

5.2Secondproofandcanonicalrepresentations...... 5-3

5.3Non-smoothpoints-asketchoffurtherresults...... 5-6

iii Bibliography

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